Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions

A. I. Ahmed; M. S. Al-Sharif; M. S. Salim; T. A. Al-Ahmary; A. I. Ahmed; M. S. Al-Sharif; M. S. Salim; T. A. Al-Ahmary

doi:10.3934/math.2022960

AIMS Mathematics

2022, Volume 7, Issue 9: 17418-17443. doi: 10.3934/math.2022960

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Research article

Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions

1.
Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut, Egypt
2.
Department of Mathematics, College of Science, King Khalid University, Abha, Saudi Arabia

Received: 05 May 2022 Revised: 04 July 2022 Accepted: 18 July 2022 Published: 28 July 2022
MSC : 34H05, 49M05, 65K10, 65L60

In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.
- fractional variational problems,
- fractional optimal control problems,
- fractional-order Chelyshkov functions,
- operational matrix,
- Lagrange multiplier method
Citation: A. I. Ahmed, M. S. Al-Sharif, M. S. Salim, T. A. Al-Ahmary. Numerical solution of fractional variational and optimal control problems via fractional-order Chelyshkov functions[J]. AIMS Mathematics, 2022, 7(9): 17418-17443. doi: 10.3934/math.2022960

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Abstract

In this paper, we present a new numerical method based on the fractional-order Chelyshkov functions (FCHFs) for solving fractional variational problems (FVPs) and fractional optimal control problems (FOCPs). The fractional derivatives are considered in the Caputo sense. The operational matrix of fractional integral for FCHFs, together with the Lagrange multiplier method, are used to reduce the fractional optimization problem into a system of algebraic equations. Some results concerning the approximation errors are discussed and the convergence of the presented method is also demonstrated. The performance of the introduced method is tested through several examples. Some comparisons with recent numerical methods are introduced to show the accuracy and effectiveness of the presented method.

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